Testing Geometric Convexity

We consider the problem of determining whether a given set S in Rn is approximately convex, i.e., if there is a convex set K ∈ Rn such that the volume of their symmetric difference is at most vol(S) for some given . When the set is presented only by a membership oracle and a random oracle, we show that the problem can be solved with high probability using poly(n)(c/ )n oracle calls and computation time. We complement this result with an exponential lower bound for the natural algorithm that tests convexity along “random” lines. We conjecture that a simple 2-dimensional version of this algorithm has polynomial complexity.

• Type: Conference
• Year: 2004
• Where: FSTTCS
• Authors: Luis Rademacher, Santosh Vempala
• Keywords: )n Oracle Calls, Convex Set, Exponential Lower Bound, FSTTCS 2004